Variational discrete variable representation

In developing a pseudospectral transform between a nondirect product basis of spherical harmonics and a direct product grid, Corey and Lemoine [J. Chem. Phys. 97, 4115 (1992)] generalized the Fourier method of Kosloff and the discrete variable representation (DVR) of Light by introducing more grid points than spectral basis functions. Assuming that the potential energy matrix is diagonal on the grid destroys the variational principle in the Fourier and DVR methods. In the present article we (1) demonstrate that the extra grid points in the generalized discrete variable representation (GDVR) act as dealiasing functions that restore the variational principle and make a pseudospectral calculation equivalent to a purely spectral one, (2) describe the general principles for extending the GDVR to other nondirect product spectral bases of orthogonal polynomials, and (3) establish the relation between the GDVR and the least squares method exploited in the pseudospectral electronic structure and adiabatic pseudospectral bound state calculations of Friesner and collaborators.